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I want to learn more about calculus on manifolds and Riemann Geometry. I have been reading the book Geometry, Topology and Physics by Nakahara. But I find that it is difficult to read due to the lack of theorems and proofs.

I would like to find a book that has a clear structure and is on a level suitable for a Ph.D student in Mathematics.

Thanks in advance!

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  • $\begingroup$ I think Spivak's series, "A Comprehensive Introduction to Differential Geometry" is pretty well regarded. I believe volume 2 covers a lot of Riemannian geometry although it gets introduced in volume 1. There are shorter, more succinct books, but these volumes flesh out the details, which it sounds like you want. $\endgroup$
    – Louis
    Commented Sep 20, 2013 at 22:11
  • $\begingroup$ It is worth mentioning John M. Lee's Introduction to Smooth Manifolds is a recent and popular text with plenty of proofs, theorems and careful definitions. I enjoyed Conlon for a quick read and the big picture. Finally, Burns and Gidea does a nice job of motivating some Riemannian geometry. It makes considerable effort to motivate some definitions which are given without much discussion elsewhere. The discussion of sectional curvatures was nice, and Burns and Gidea also has a dynamical systems bent for those interested. $\endgroup$
    – James S. Cook
    Commented Sep 21, 2013 at 0:52

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Riemannian Geometry by M. do Carmo is a great book that takes a variational approach, but I feel it is somewhat old-fashioned.

Riemannian Geometry by Peter Petersen is another great book that takes a very modern approach and contains some specialized topics like convergence theory.

Geometric Analysis by Peter Li is a great book that focuses on the PDE aspects of the theory, and it is based on notes freely available on his website so you can get a taste of it.

Riemannian Geometry by W. Klingenberg is another reference that has helped me out a lot over the years.

If you want to go back in time and learn from a master, consider

Differential Geometry in the Large by Heinz Hopf. It is still worth a read!

People always mention the M. Spivak books when users ask for Geometry references, and they are wonderful, well written, great books. But I never find myself referencing this book when doing research, even though it is on my shelf. I personally found it to be a little long winded, although there is some nice stuff in volume 5 about isometric immersions.

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  • $\begingroup$ You're welcome! It goes (almost) without saying that these represent just some of my personal preferences, so by nature there is some subjectivity here. $\endgroup$
    – treble
    Commented Sep 20, 2013 at 22:26
  • $\begingroup$ Jack Lee's Riemannian Geometry book is also quite good (but doesn't cover as much as do Carmo or Petersen). $\endgroup$
    – user641
    Commented Sep 21, 2013 at 0:03
  • $\begingroup$ Hi, do you know if M. do Carmo treats both Levi Civita connection and Ricci flow? (I'm trying to find a good reference treating this stuff). $\endgroup$
    – user8469759
    Commented Jan 15, 2019 at 13:55

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